Expected Bond Returns
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1 994 Expected Bond Returns 994 Yield Curve: February to April Yield Curve: February April, April 994 Chris Telmer May 03 Nominal Yields (% CCAR) February Maturity (Q) / 8 / 8 Greenspan Uses the Expectations Hypothesis 994 In early February, we thought long-term rates would move a little higher as we tightened. The sharp jump in [long] rates that occurred appeared to reflect the dramatic rise in market expectations of economic growth and associated concerns about possible inflation pressures....alan Greenspan, May 7, Yield Curve: May 004 to June Nominal Yields (% CCAR) Yield Curve: May 004 June 6 5 June 4 3 May Maturity (Q) 3 / 8 4 / 8
2 Greenspan s 994 The Greenspan. Or, Alan Greenspan (finally!) Rejects the Expectations Hypothesis. Long-term interest rates have trended lower in recent months even as the Federal Reserve has raised the level of the target federal funds rate by 50 basis points. Historically, even distant forward rates have tended to rise in association with monetary policy tightening.... For the moment, the broadly unanticipated behavior of world bond markets remains a conundrum. 994 Nominal Yields Across Time and Maturity Yields (% CCAR) Nominal Yields (970: to :4) Alan Greenspan, February 6, Maturity (Q) / 8 6 / 8 USD-denominated price of an n-period zero-coupon bond: Continuously compounded yield or spot interest rate: b n t y n t = n logbn t Forward interest rates: f n t = log(b n t /b n+ t ) Yields are averages of forward rates. n yt n = n i=0 Short rate defined as i t. By definition, i t = y t = f 0 t. f i t Holding-period returns (HPRs). The continuously-compounded HPR on an n-period bond, between t and t+ is defined as: h n t+ logb n t+ logb n t 7 / 8 8 / 8
3 Forward interest rates Picture Yields on zero-coupon bonds are usually called spot interest rates. They are often computed using the yields on coupon bonds ( bootstrapping ). What is the forward interest rate? Above, it looks like the interest rate that applies between one-year-from-now and two-years-from-now. Definition: But note that it is essentially a definition, once we know i t and y t. f t logb t logb t = i t +yt = yt = ( it +f ) t Or is it...? On the next page we show that it is actually a market interest rate. 9 / 8 0 / 8 (continued) Consider a cash inflow of USD 00, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. You can arrange to lend these USD 00 at the forward interest rate! Short sell 00 one-year bonds and invest the proceeds in two-year bonds. The short sale generates 00 b t dollars, which will buy 00 b t /b t two-year bonds. In one year we must pay the short-sale liability of USD 00. This is exactly offset by the cash inflow. In two years we receive USD 00 b t /b t The definition of the forward rate implies that: The Expectations Hypothesis for the Term Structure of Interest Rates b t b t = /(+i t) /((+i t)(+f t )) = (+f t ) Therefore, our portfolio generates USD 00 (+ft ) in two years. This means that ft is the implicit interest rate at which one can borrow (lend) between one year from now and two years from now. / 8 / 8
4 The Expectations Hypothesis The expected return on any bond strategy is the same* Consider the two-period HPR on three-period bond: Example: HPR on -year zero minus HPR on one-year zero: h t+ h t+ = logb t+ logb t ( 0 logb ) t = logb t+ + ( logb t logb ) t = logb t+ +ft = E t (h t+ h t+ ) = 0 so that f t = E t logb t+ h 3 t,t+ ( logb t+ logb 3 t) / Subtract it from the two-period HPR on two-period bond: h t,t+ ( 0 logb t) / Use the same arithmetic as above to demonstrate that f t = E t y t+ = E t y t+ = E t i t+ Therefore, in general, *By return we will mean continuously-compounded return. Note that, if we also were to consider discretely-compounded returns, the various ways of writing the would be inconsistent with each other, and inconsistent with what s above. The reason is basically Jensen s inequality. In FOREX this gets referred to as Siegel s paradox: if the holds in USD units, it can t in GBP units. Sticking with continuous compounding avoids much of this. Quantitatively none of this amounts to much. There s an old paper by Cox-Ingersoll-Ross that spells-out all of this. f n t = E t y t+n The says that forward interest rates are expected future short rates. 3 / 8 4 / 8 the = E(future spots) Unconditional test: average HPRs equal by maturity. ft n = E t yt+n Expected HPRs equal for all n: E t h n t+ = E t h m t+ = yt E t h n t+ = E t h m t+ = y t = Eh n t+ = Eh m t+ = y t Long yields are expected averages of future short rates: yt n ( = E t y t+ +yt yt+n ) 5 / 8 Source: Cochrane, John, New facts in finance, Economic Perspectives, Chicago Fed, April / 8
5 (continued) Results: Panel A Conditional tests: f n t = E t yt+n = ft n yt ( = E t y t+n yt ) = ft n yt = ( yt+n yt) εt+n where ε t+n is defined as the forecast error, s.t., yt+n yt ( E t y t+n yt) +εt+n This suggests the regression: y t+n y t = a+b ( f n t y t) +residuals Source: Cochrane, John, New facts in finance, Economic Perspectives, Chicago Fed, April 999. The null is a = 0, b = 7 / 8 8 / 8 Regression Results: Panel B If the is violated then, necessarily, there are excess expected returns available. The only issue is how much risk? A different (complementary) regression is useful. Recall, the says that, for all n, E t h n t+ = y t So, how about, for n >, h n t+ y t The null is a = 0, b = 0. = a+b ( f n t y t) +residuals If b 0 then the n-period forward premium predicts the excess HPR on the n-period bond. Source: Cochrane, John, New facts in finance, Economic Perspectives, Chicago Fed, April / 8 0 / 8
6 Fixed Income of the Evidence For n =,3,..., run the regression, get estimates of a and b, â and ˆb h n t+ y t Carry trade: at each date (month), t: If â+ˆb ( ft n If â+ˆb ( ft n = a+b ( f n t y t) +residuals yt ) ) y t > 0 borrow short, invest long < 0 borrow long, invest short Unconditional: holds (pretty much) It doesn t matter if you buy-and-hold n year bonds or m years bonds. Conditional: violated There are times when the yield curve is steep enough to justify borrowing short and investing long There are other times when the reverse is true If you average across these times, you get the unconditional evidence An analogous result is true for currencies Out-of-sample backtesting, etc., important Bottom line: there are risk premiums inherent in the behavior of the term structure / 8 / 8 (continued) (continued) Here s another clever way to think about the conditional evidence: How different are realized short rates from predicted short rates Suppose that the yield curve is strongly upward-sloping The idea of the is that if you invest long, future short rates will increase. This will depress the HPRs on your long bonds and make the (implicit alternative) portfolio of rolled-over shorts look good. Your (apparent) excess yield gets gobbled up. Does this happen? Yes and no... Source: Cochrane, John, New facts in finance, Economic Perspectives, Chicago Fed, April / 8 4 / 8
7 Discussion of Fama-Bliss Evidence in favor of excess expected returns on certain bond portfolios is evidence against risk neutrality One uses an arbitrage-free (aka risk neutral) term structure model for asset allocation and/or risk management at one s peril. The evidence indicates that m t is stochastic and that we must model it to do a serious job of fixed-income asset allocation and risk management Discussion of Fama-Bliss assignment Approaches: Affine term structure models Factor models 5 / 8 6 / 8 What is the? What is the? What is the? People sometimes think that the is specific to bonds and currencies. This isn t quite right. The is just the statement that expected returns are equal across different assets. Bonds and currencies just happen to be fixed-income, so things are nice and clean. But we could just as easily write: E t q j t+ +δj t+ q j t = E t q k t+ +δk t+ q k t for two stocks, j and k. This would be the for equities. If there are forwards on stocks the same sort of forwards equal expected future spots would pop out. 7 / 8 8 / 8
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